Calculus Chapter 1: Functions, Limits, and Continuity – Study Notes

Functions

Definition and Domain of Functions

A function is a relation that assigns each element in the domain to exactly one element in the range. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Finding the Domain: Exclude values that make denominators zero or result in negative values under even roots.

• Example: For $f(x) = \frac{2x+1}{x^2-x-2}$, the domain excludes $x=2$ and $x=-1$ because they make the denominator zero.

Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x, & x < 0 \\ x^2, & 0 \leq x < 1 \\ 1, & x \geq 1 \end{cases}$

• Graphing piecewise functions requires plotting each segment over its specified interval.

Increasing and Decreasing Functions

A function is increasing on an interval if $f(x_1) < f(x_2)$ for $x_1 < x_2$; decreasing if $f(x_1) > f(x_2)$ for $x_1 < x_2$.

• Intervals of increase/decrease can be found by analyzing the function's graph or derivative.

Even and Odd Functions

Even functions satisfy $f(-x) = f(x)$ and are symmetric about the y-axis. Odd functions satisfy $f(-x) = -f(x)$ and are symmetric about the origin.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Composite Functions

A composite function is formed by applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• Example: If $f(x) = x^2$ and $g(x) = \sqrt{x}$, then $(f \circ g)(x) = f(g(x)) = (\sqrt{x})^2 = x$.

Shifting and Reflecting Graphs

Functions can be shifted vertically and horizontally, or reflected about axes.

• Vertical Shift: $y = f(x) + k$ shifts up if $k > 0$, down if $k < 0$.

• Horizontal Shift: $y = f(x + h)$ shifts left if $h > 0$, right if $h < 0$.

• Reflection: $y = -f(x)$ reflects about the x-axis; $y = f(-x)$ reflects about the y-axis.

Trigonometric Functions

Basic Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Periodic Functions

Trigonometric functions are periodic, repeating their values in regular intervals.

• Period of $\sin x$, $\cos x$: $2\pi$

• Period of $\tan x$, $\cot x$: $\pi$

Trigonometric Identities

Identities are equations true for all values in the domain.

• $\sin^2 x + \cos^2 x = 1$

• $1 - \cos 2x = 2\sin^2 x$

• $\cos 2x = 2\cos^2 x - 1$

Graphs of Trigonometric Functions

Each trigonometric function has a distinct graph, showing periodic behavior and symmetry.

• Example: The graph of $\sin x$ oscillates between -1 and 1 with period $2\pi$.

Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Properties: Exponential functions grow rapidly and are always positive for $a > 0$.

• Example: $f(x) = 3^x$

Rules of Exponents

• $a^x \cdot a^y = a^{x+y}$

• $\frac{a^x}{a^y} = a^{x-y}$

• $(a^x)^y = a^{xy}$

• $(ab)^x = a^x b^x$

Logarithmic Functions

A logarithmic function is the inverse of an exponential function: $y = \log_a x$ is equivalent to $a^y = x$.

• Natural logarithm: $\ln x = \log_e x$

• Properties: $\ln(ab) = \ln a + \ln b$, $\ln(a/b) = \ln a - \ln b$, $\ln(a^b) = b \ln a$

Limits

Definition of a Limit

The limit of $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x) = L$

• Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$

Limit Laws

Limits can be evaluated using algebraic rules:

• $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$

• $\lim_{x \to c} [f(x)g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

• $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, if $\lim_{x \to c} g(x) \neq 0$

Evaluating Limits

Limits can be evaluated by substitution, factoring, rationalizing, or using special limit theorems.

• Example: $\lim_{x \to 1} \frac{\sqrt{x+4} - 2}{x - 1} = \frac{1}{4}$

One-Sided Limits

One-sided limits consider the behavior of $f(x)$ as $x$ approaches $c$ from the left ($x \to c^-$) or right ($x \to c^+$).

• Notation: $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$

Limits Involving Infinity

Limits can approach infinity or negative infinity, indicating unbounded behavior.

• Horizontal Asymptotes: $\lim_{x \to \infty} f(x) = L$

• Vertical Asymptotes: $\lim_{x \to a} f(x) = \infty$

Sandwich (Squeeze) Theorem

If $f(x) \leq g(x) \leq h(x)$ near $x=a$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

Continuity

Definition of Continuity

A function $f(x)$ is continuous at $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Properties of Continuous Functions

If $f$ and $g$ are continuous at $a$, then so are $f+g$, $f-g$, $fg$, $f/g$ (if $g(a) \neq 0$), and $f(g(x))$.

Types of Discontinuity

• Removable discontinuity: The limit exists but $f(a)$ is not defined or not equal to the limit.

• Jump discontinuity: Left and right limits exist but are not equal.

• Infinite discontinuity: The function approaches infinity near $a$.

Intermediate Value Theorem

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $[a, b]$ such that $f(c) = N$.

Continuous Extension

A function not defined at a point can sometimes be extended to be continuous at that point by defining $f(a)$ as the limit as $x \to a$.

Summary Table: Properties of Functions

Property	Definition	Example
Domain	Set of all $x$ for which $f(x)$ is defined	$f(x) = \frac{1}{x}$, domain: $x \neq 0$
Even Function	$f(-x) = f(x)$	$f(x) = x^2$
Odd Function	$f(-x) = -f(x)$	$f(x) = x^3$
Composite Function	$(f \circ g)(x) = f(g(x))$	$f(x) = x^2$, $g(x) = \sqrt{x}$, $f(g(x)) = x$
Continuous at $a$	$\lim_{x \to a} f(x) = f(a)$	$f(x) = x^2$ at $x=1$

Key Formulas

• $\sin^2 x + \cos^2 x = 1$

• $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ (definition of derivative, additional info: not covered in detail in these notes)

• $\ln(ab) = \ln a + \ln b$

• $a^x \cdot a^y = a^{x+y}$

Additional info: These notes cover the foundational topics of Chapter 1 (Functions), Chapter 2 (Limits and Continuity), and include essential trigonometric, exponential, and logarithmic functions, as well as their properties and graphs. The content is suitable for college-level Calculus students preparing for exams or seeking a structured summary of the first chapters of a Calculus textbook.